Cokernel

<h2 id="formal-definition">Formal definition</h2>
One can define the cokernel in the general framework of <a href="/facts/Category_theory/6zS3DXPa">category theory</a>. In order for the definition to make sense the category in question must have <a href="/facts/Zero_morphism/Klk9tyde">zero morphisms</a>. The cokernel of a <a href="/facts/Morphism/Kdpuyz3S">morphism</a> f : X → Y is defined as the <a href="/facts/Coequalizer/UjkYMAfm">coequalizer</a> of f and the zero morphism 0XY : X → Y.
Explicitly, this means the following. The cokernel of f : X → Y is an object Q together with a morphism q : Y → Q such that the diagram

<a href="/facts/Commutative_diagram/bdG8xuer">commutes</a>. Moreover, the morphism q must be <a href="/facts/Universal_property/5XJBYMNz">universal</a> for this diagram, i.e. any other such q′ : Y → Q′ can be obtained by composing q with a unique morphism u : Q → Q′:

As with all universal constructions the cokernel, if it exists, is unique <a href="/facts/Up_to/ydz4ztzX">up to</a> a unique <a href="/facts/Isomorphism/pj8KgkxU">isomorphism</a>, or more precisely: if q : Y → Q and q′ : Y → Q′ are two cokernels of f : X → Y, then there exists a unique isomorphism u : Q → Q′ with q' = u q.
Like all coequalizers, the cokernel q : Y → Q is necessarily an <a href="/facts/Epimorphism/IrzonI01">epimorphism</a>. Conversely an epimorphism is called <a href="/facts/Normal_morphism/vSYyev04">normal</a> (or conormal) if it is the cokernel of some morphism. A category is called conormal if every epimorphism is normal (e.g. the <a href="/facts/Category_of_groups/sxSd9Kbc">category of groups</a> is conormal).

<h3>Examples</h3>
In the <a href="/facts/Category_of_groups/sxSd9Kbc">category of groups</a>, the cokernel of a <a href="/facts/Group_homomorphism/sDguaNYs">group homomorphism</a> f : G → H is the <a href="/facts/Quotient_group/HlwpHtjb">quotient</a> of H by the <a href="/facts/Normal_closure_(group_theory)/k2lJahCW">normal closure</a> of the image of f. In the case of <a href="/facts/Abelian_group/FfWwmx4R">abelian groups</a>, since every <a href="/facts/Subgroup/r91mBgpa">subgroup</a> is normal, the cokernel is just H <a href="/facts/Ideal_(ring_theory)/vCvytduX">modulo</a> the image of f:

coker
        ⁡
        (
        f
        )
        =
        H
        
          /
        
        im
        ⁡
        (
        f
        )
        .
      
    
    {\displaystyle \operatorname {coker} (f)=H/\operatorname {im} (f).}

<h3>Special cases</h3>
In a <a href="/facts/Preadditive_category/l308bN5A">preadditive category</a>, it makes sense to add and subtract morphisms. In such a category, the <a href="/facts/Coequalizer/UjkYMAfm">coequalizer</a> of two morphisms f and g (if it exists) is just the cokernel of their difference:

coeq
        ⁡
        (
        f
        ,
        g
        )
        =
        coker
        ⁡
        (
        g
        −
        f
        )
        .
      
    
    {\displaystyle \operatorname {coeq} (f,g)=\operatorname {coker} (g-f).}

In an <a href="/facts/Abelian_category/4WN473Yc">abelian category</a> (a special kind of preadditive category) the <a href="/facts/Image_(category_theory)/r82bXCQc">image</a> and <a href="/facts/Coimage/khlveI8X">coimage</a> of a morphism f are given by

im
                ⁡
                (
                f
                )
              
              
                
                =
                ker
                ⁡
                (
                coker
                ⁡
                f
                )
                ,
              
            
            
              
                coim
                ⁡
                (
                f
                )
              
              
                
                =
                coker
                ⁡
                (
                ker
                ⁡
                f
                )
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\operatorname {im} (f)&=\ker(\operatorname {coker} f),\\\operatorname {coim} (f)&=\operatorname {coker} (\ker f).\end{aligned}}}

In particular, every abelian category is normal (and conormal as well). That is, every <a href="/facts/Monomorphism/Q9BrtDTs">monomorphism</a> m can be written as the kernel of some morphism. Specifically, m is the kernel of its own cokernel:

m
        =
        ker
        ⁡
        (
        coker
        ⁡
        (
        m
        )
        )
      
    
    {\displaystyle m=\ker(\operatorname {coker} (m))}

<h2 id="intuition">Intuition</h2>
The cokernel can be thought of as the space of constraints that an equation must satisfy, as the space of obstructions, just as the <a href="/facts/Kernel_(algebra)/GGSpUPMJ">kernel</a> is the space of solutions.
Formally, one may connect the kernel and the cokernel of a map T: V → W by the <a href="/facts/Exact_sequence/leUJnwJb">exact sequence</a>

0
        →
        ker
        ⁡
        T
        →
        V
        
          
            ⟶
            T
          
        
        W
        →
        coker
        ⁡
        T
        →
        0.
      
    
    {\displaystyle 0\to \ker T\to V{\overset {T}{\longrightarrow }}W\to \operatorname {coker} T\to 0.}

These can be interpreted thus: given a linear equation T(v) = w to solve,

<ul><li>the kernel is the space of solutions to the homogeneous equation T(v) = 0, and its dimension is the number of degrees of freedom in solutions to T(v) = w, if they exist;</li>
<li>the cokernel is the space of constraints on w that must be satisfied if the equation is to have a solution, and its dimension is the number of independent constraints that must be satisfied for the equation to have a solution.</li></ul>
The dimension of the cokernel plus the dimension of the image (the rank) add up to the dimension of the target space, as the dimension of the quotient space W / T(V) is simply the dimension of the space minus the dimension of the image.
As a simple example, consider the map T: R2 → R2, given by T(x, y) = (0, y). Then for an equation T(x, y) = (a, b) to have a solution, we must have a = 0 (one constraint), and in that case the solution space is (x, b), or equivalently, (0, b) + (x, 0), (one degree of freedom). The kernel may be expressed as the subspace (x, 0) ⊆ V: the value of x is the freedom in a solution. The cokernel may be expressed via the real valued map W: (a, b) → (a): given a vector (a, b), the value of a is the obstruction to there being a solution.
Additionally, the cokernel can be thought of as something that "detects" <a href="/facts/Surjection/KezhLpCb">surjections</a> in the same way that the kernel "detects" <a href="/facts/Injection_(mathematics)/5ES6tqf5">injections</a>. A map is injective if and only if its kernel is trivial, and a map is surjective if and only if its cokernel is trivial, or in other words, if W = im(T).

<ul><li><a href="/facts/Saunders_Mac_Lane/rjgUqfkA">Saunders Mac Lane</a>: <a href="/facts/Categories_for_the_Working_Mathematician/SLQDat9B">Categories for the Working Mathematician</a>, Second Edition, 1978, p. 64</li>
<li><a href="/facts/Emily_Riehl/sPtNtfrP">Emily Riehl</a>: <a href="http://www.math.jhu.edu/~eriehl/context.pdf#page=100">Category Theory in Context</a>, <a href="https://store.doverpublications.com/048680903x.html">Aurora Modern Math Originals</a>, 2014, p. 82, p. 139 footnote 8.</li></ul>

Cokernel open-in-new

Cokernel